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Basic Inequalities for Real Hypersurfaces in Some Grassmannians

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Abstract

In this paper, we obtain some basic inequalities for real hypersurfaces of complex two-plane Grassmannians and complex hyperbolic two-plane Grassmannians.

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References

  1. Al-Solamy, F.R., Bansal, P.: Geometry of Chen invariants in statistical warped product manifolds. Int. J. Geom. Methods Mod. Phys. 17(06), 2050081 (2020)

    Article  MathSciNet  Google Scholar 

  2. Alodan, H., Chen, B.-Y., Deshmukh, S., Vîlcu, G.-E.: A generalized Wintgen inequality for quaternionic CR-submanifolds. Revista De Ls Real Academia De Cincias Exactas Fisicas Y Naturales Series A Mathematics 114(3), 1–14 (2020)

    MathSciNet  MATH  Google Scholar 

  3. Beem, J., Ehrlich, P., Powell, T.G.: Warped product manifolds in relativity. In: Selected Studies: A Volume Dedicated to the Memory of Albert Einstein, pp. 41–56 (1982)

  4. Beem, J.K.: Global lorentzian geometry. Routledge, London (2017)

    Book  Google Scholar 

  5. Berndt, J.: Riemannian geometry of complex two-plane Grassmannians. Rend. Sem. Mat. Univ. Politec. Torino 55(1), 19–83 (1997)

    MathSciNet  MATH  Google Scholar 

  6. Berndt, J., Suh, Y.J.: Hypersurfaces in noncompact complex Grassmannians of rank two. Int. J. Math. 23(10), 1250103 (2012)

    Article  MathSciNet  Google Scholar 

  7. Bishop, R.L., O’Neill, B.: Manifolds of negative curvature. Trans. Am. Math. Soc. 145, 1–49 (1969)

    Article  MathSciNet  Google Scholar 

  8. Chen, B.-Y.: Some pinching and classification theorems for minimal submanifolds. Archiv der Mathematik 60(6), 568–578 (1993)

    Article  MathSciNet  Google Scholar 

  9. Chen, B.-Y.: Mean curvature and shape operator of isometric immersions in real-space-forms. Glasg. Math. J. 38(1), 87–97 (1996)

    Article  MathSciNet  Google Scholar 

  10. Chen, B.-Y.: On isometric minimal immersions from warped products into real space forms. Proc. Edinb. Math. Soc. 45(3), 579–587 (2002)

    Article  MathSciNet  Google Scholar 

  11. Chen, B.-Y.: Pseudo-Riemannian geometry, \(\delta \)-invariants and applications. World Scientific, Singapore (2011)

    Book  Google Scholar 

  12. Chen, B.-Y., Mihai, A., Mihai, I.: A Chen first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature. Results Math. 74(4), 165 (2019)

    Article  MathSciNet  Google Scholar 

  13. Chen, X.: Real hypersurfaces of complex two-plane Grassmannians with certain parallel conditions. J. Geom. 108(3), 1157–1174 (2017)

    Article  MathSciNet  Google Scholar 

  14. de Dios Pérez, J.: Real hypersurfaces of quaternionic projective space satisfying \( \nabla _{U_i} A= 0 \). J. Geom. 49(1–2), 166–177 (1994)

    Article  MathSciNet  Google Scholar 

  15. de Dios Pérez, J., Suh, Y.J., Woo, C.: Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting shape operator. Open Math. 1(open-issue) (2015)

  16. Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time, vol. 1. Cambridge University Press, Cambridge (1973)

    Book  Google Scholar 

  17. Hwang, D.H., Pak, E., Woo, C.: Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting restricted normal Jacobi operators. Czechoslov. Math. J. 67(4), 989–1004 (2017)

    Article  MathSciNet  Google Scholar 

  18. Ianuş, S., Mazzocco, R., Vîlcu, G.E.: Riemannian submersions from quaternionic manifolds. Acta Applicandae Mathematicae 104(1), 83–89 (2008)

    Article  MathSciNet  Google Scholar 

  19. Kim, B.H., Lee, H., Pak, E.: Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster Reeb-parallel structure Jacobi operator. Kyungpook Math. J. 59(3), 525–535 (2019)

    MathSciNet  MATH  Google Scholar 

  20. Kim, G.J., Suh, Y.J.: Real hypersurfaces in complex hyperbolic two-plane Grassmannians with Reeb invariant Ricci tensor. Differ. Geom. Appl. 47, 14–25 (2016)

    Article  MathSciNet  Google Scholar 

  21. Kimura, M.: Real hypersurfaces of a complex projective space. Bull. Aust. Math. Soc. 33(3), 383–387 (1986)

    Article  MathSciNet  Google Scholar 

  22. Kimura, M., Maeda, S.: On real hypersurfaces of a complex projective space II. Tsukuba J. Math. 15(2), 547–561 (1991)

    Article  MathSciNet  Google Scholar 

  23. Macsim, G., Mihai, A., Mihai, I.: \(\delta \) (2, 2)-invariant for Lagrangian submanifolds in quaternionic space forms. Mathematics 8(4), 480 (2020)

    Article  Google Scholar 

  24. Mihai, A., Mihai, I.: The \(\delta \) (2, 2)-invariant on statistical submanifolds in Hessian manifolds of constant Hessian curvature. Entropy 22(2), 164 (2020)

    Article  MathSciNet  Google Scholar 

  25. Mihai, A., Özgür, C.: Chen inequalities for submanifolds of complex space forms and Sasakian space forms endowed with semi-symmetric metric connections. Rocky Mt. J. Math. 41, 1653–1673 (2011)

    Article  MathSciNet  Google Scholar 

  26. Naaz Siddiqui, A., Ali, A., Alkhaldi, A.H.: Chen optimal inequalities of CR-warped products of generalized Sasakian space form. Journal of Taibah University for. Science 14(1), 322–330 (2020)

    Google Scholar 

  27. Pak, E., Suh, Y.J.: \(d\perp \)-parallel normal Jacobi operators for Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster connection. Adv. Geom. 20(2), 163–168 (2020)

    Article  MathSciNet  Google Scholar 

  28. Park, K.-S.: Inequalities for the Casorati curvatures of real hypersurfaces in some Grassmannians. Taiwan. J. Math. 22(1), 63–77 (2018)

    Article  MathSciNet  Google Scholar 

  29. Park, K.-S.: On the estimates of warping function on isometric immersions. U.P.B. Sci. Bull. Ser. A 82(1), 13–26 (2020)

    MathSciNet  Google Scholar 

  30. Suh, Y.J.: Real hypersurfaces in complex two-plane Grassmannians with parallel shape operator. Bull. Aust. Math. Soc. 67(3), 493–502 (2003)

    Article  MathSciNet  Google Scholar 

  31. Suh, Y.J.: Generalized Killing Ricci tensor for real hypersurfaces in complex two-plane Grassmannians. J. Geom. Phys. 159, 103799 (2021)

    Article  MathSciNet  Google Scholar 

  32. Suh, Y.J.: Harmonic curvature for real hypersurfaces in complex hyperbolic two plane Grassmannians. J. Geom. Phys. 161, 103829 (2020)

    Article  MathSciNet  Google Scholar 

  33. Uddin, S., Chen, B.-Y., Al-Solamy, F.R.: Warped product bi-slant immersions in Kaehler manifolds. Mediterr. J. Math. 14(2), 95 (2017)

    Article  MathSciNet  Google Scholar 

  34. Vîlcu, A.-D., Vîlcu, G.-E.: Statistical manifolds with almost quaternionic structures and quaternionic kähler-like statistical submersions. Entropy 17(9), 6213–6228 (2015)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are very thankful to the reviewers for their valuable comments and suggestions which helped a lot to improve the quality of this paper.

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Correspondence to Mehraj Ahmad Lone.

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Lone, M.S., Lone, M.A. Basic Inequalities for Real Hypersurfaces in Some Grassmannians. Mediterr. J. Math. 18, 203 (2021). https://doi.org/10.1007/s00009-021-01863-w

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  • DOI: https://doi.org/10.1007/s00009-021-01863-w

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